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This video is provided as supplementary material for courses taught a Howard Community College and,
in this video I'm going to talk about the probability of the complement of
an event.
So I'll start with this example.
An airline reports that 94% of its flights arrive on time.
What is the probability that will be late?
So let's call event of arriving on time,
"A".
So I can say that the probability
of A,
the probability of arriving on time,
is 94%.
I want to find the probability
that a flight will be late.
So let's call that the probability of not arriving on time,
or the probability not A.
The probability of not A.
Now there are only two things that can happen: a plane can arrive on time
or it cannot arrive on time.
Since those are the only two events that I can have, if I add those two events
together, I'm going to get 100%.
So the probability of A
plus the probability
of
not A
is equal to 100%.
If I take that equation and subtract
the probability of A from both sides,
I'll find the probability of no A.
So the probability
of not A
is equal to
100%
minus
the probability of A.
Well, the probability of A is 94%. So the probability of not A equals
100%
minus
94%.
If I subtract 94% from 100%
I get 6%.
So the probability of not A, the probability of
a plane arriving late,
is 6%.
Now when we talk about A
and not A,
we talk about these as being complements of each other.
In other words, if A is an event,
its complement,
the event not happening,
is not A.
We can write that as 'not A'.
If we want to use a different notation,
I can just write anA
with a bar over it.
So an 'A' with a bar over it means the complement of A.
So I'll change these other
places where I've written 'not' and just put a bar
over the A.
And so the general way you're going to find the complement of an event,
the way you're going to find what happens when an event doesn't occur, the probability of
it,
is going to be
to subtract the probability of the event occurring
from 100%, if we're doing with a percent.
If it's not a percent, just subtracted it from one.
Okay, let's do a couple more examples.
In this example I've got a spinner
and a circle. The circle is divided into five sections. The sections are labeled
A, B, C, D and E.
And the probability
of the spinner
landing in section A when I spin it
is 0.23. So I've got that probability as a decimal,
23 hundredths.
I want to find the probability
of it not landing in section A.
In other words, the probability of not A,
or
the complement of A,
A with a bar over it.
So all I have to do is realize
that the probability of the A
plus the probability of A not happening, the probability of
A-complement,
or the complement of A,
is 1.
If I subtract the probability of A from both sides,
I'm going to have the probability
of the complement of A
equaling 1
minus
the probability of A.
The probability of A was 0.23.
When I subtract 1
minus 0.23,
I'm going to end up with
0.77.
0.77.
So that's the probability
of the spinner not landing in section A,
the probability of the complement of A.
Let's look at one more.
In a bag containing 29 marbles,
5 of the marbles are red and 2 are green.
What is the probability of randomly selecting a marble
which is neither red
nor green?
Okay. So
the first thing we have to do is find the probability of getting a red
marble and the probability of getting a green marble.
So the probability of red, we'll call that the probability of R,
is
is going to be 5
over
29,
because there are 5 different ways I can get a red marble
and there are 29
total marbles I could select.
The probability of getting a green marble
is going to be 2 over 29.
There only 2 green marbles
out of the 29 that I could choose.
If I add these two probabilities together,
I'll get the probability
getting either a red
or a green marble.
And I'll show that as the probability of the union
of red
and green.
Add 5/29 plus 2/29,
I just have to add 5 and 2. That 7
over 29.
So the probability of getting either red or green is 7/29.
The complement of getting red or green
would be the probability of getting neither red nor green.
I can write that
as the probability of the complement
of the union
of red or green.
And the way to find that is going to be to subtract the probability of getting
red or green
from one.
So instead of 1,
I'll use 29/29, to make the subtraction easy.
I'll subtract 7/29 from that.
When I subtract 29 minus 7 I get 22.
So the probability
of getting neither red
nor green,
the probability of the complement of red or green,
is going to be
22
over 29.
So the basic principle for all of the examples I had was this:
Take the probability of something happening, the probability that you know,
and just subtract that
from
one, if it's a decimal or a fraction,
or from 100% if you're dealing with a percent, and that will give you
the probability of the event not happening,
the probability of the complement
of the event. And that's all there is to it.
Take care,
I'll see you next time.